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ATLANTA — Delta Air Lines’ transformation plan, which includes cutting annual costs by $5 billion by the end of next year, is not enough to save the struggling carrier, its chief executive said in a memo to employees that addressed renewed concerns about bankruptcy. The airline’s shares plunged.

CEO Gerald Grinstein said in Tuesday’s memo, which was obtained yesterday, that the nation’s third-largest airline is still working hard to avoid a Chapter 11 filing, but he reiterated there are risks affecting Delta’s ability to do that.

“In light of what we have accomplished together so far, there can be no doubt that Delta’s transformation plan is delivering results,” Grinstein said. “What is also clear is that is not enough.”

Grinstein, citing the impact of high fuel prices and the interest expense on the company’s massive debt, said Delta plans to expand its initiatives to cut costs, improve efficiencies and raise cash.

“Given our financial situation, there is renewed speculation about bankruptcy,” Grinstein told employees. “We have been candid about the risk that a number of factors, some of which are beyond our control, will affect our ability to avoid a Chapter 11 filing. However, we are still working to pursue an out-of-court solution, even as we face increasing financial pressures.”

Delta shares fell 40 cents, or 15 percent, to close at $2.99 yesterday.

Airline analyst Ray Neidl said Grinstein’s comments in the memo and during a conference call with investors last week suggest that Delta is preparing people for the worst, even as it tries to keep that from happening.

“The market is becoming more dubious of Delta’s chances of avoiding bankruptcy,” said Neidl, of Calyon Securities. “I know I am.”

Last week, Delta reported a $388 million second-quarter loss, pushing its red ink to nearly $10 billion since early 2001.

Some analysts have speculated that if Congress doesn’t pass meaningful pension-funding reform by fall, Delta will be forced into bankruptcy. But Grinstein has noted that the airline has other equally pressing concerns, including hefty fuel costs.

In the memo, Grinstein acknowledged that it may be puzzling to Delta employees that, despite high fuel prices, several other legacy carriers that face the same problem as Delta were able to post modest profits in the second quarter.

“What’s going on?” Grinstein said. “Simply put, it is in large part a matter of timing and competitive market challenges unique to Delta.”

Also yesterday, Delta said it is now offering customers who buy their tickets on the airline’s Web site the ability to cancel certain tickets within 24 hours without penalty if they find lower fares on another carrier or if their travel plans change.